Parabolas and curves, I've been thinking, are molded by infinite numbers of tangents, or lines, correct?

Answer 1

Many curves are like that, yes. Care for something fun? Get a piece of squared paper. Number it from 1 to 9 (or other max number) on the vertical, and from 1 to 9 on the horizontal. But, unlike a graph, they should be at opposite ends. In other words, if 1 though 9 counts up from 1 at the bottom, and 9 at the top, then the horizonal extends to the right, you should have a 1 out at the end. It looks like this:


9
8
7
6
5
4
3
2
1
. 9 8 7 6 5 4 3 2 1

Draw a straight line from 1 to 1, 2 to 2, 3 to 3, etc. Enjoy.

Theres some nifty writeups on it in the source below:

Answer 2

In fact parabolas is a path described by a moving point under certain conditions. e.g. : y^2 =4ax is the locus for right handed parabola. Infinite tangents are possible for an unbounded parabola.

Answer 3

Simply put yes. You are actually touching on a fundamental concept of calculus. The derivative.

This is not true for all functions. One example of such a function is the absolute valve of x, |x|. At x=0 the tangent depends on if you are coming from x<0 or x>0.

Answer 4

A continuous function is completely defined by the values it takes on rational numbers. This is not just any old infinity, but a countable one. Try using this to find a one-to-one mapping from the set of all continuous functions to the set of all reals.

Answer 5

Basically, yes. The tangents move (rotate) as you move along the curve, and that is the concavity concept.